Spectrally accurate fully discrete schemes for some nonlocal and nonlinear integrable PDEs via explicit formulas

2024/12/18 by Bronsard, Yvonne Alama, Chen, Xi, Dolbeault, Matthieu · 1 citation
#35Q35 #37K10 #37K15 #65M15 #65M70 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2412.13480

Abstract

We construct fully-discrete schemes for the Benjamin-Ono, Calogero-Sutherland DNLS, and cubic Szegő equations on the torus, which are exact in time with spectral accuracy in space. We prove spectral convergence for the first two equations, of order K^-s+1 in L² norm for initial data in H^s(\mathbb T), s>1, with an error constant depending linearly on the final time instead of exponentially. These schemes are based on explicit formulas, which have recently emerged in the theory of nonlinear integrable equations. Numerical simulations show the strength of the newly designed methods both at short and long time scales, thanks to the remarkable fact that the computational cost of the method is independent of the final time. These schemes open doors for the understanding of the long-time dynamics of integrable equations.

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• Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation

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